Spectral flows associated to flux tubes
When a flux quantum is pushed through a gapped two-dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin’s Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic signature, while for particle-hole symmetric systems it leads to a criterion for the existence of zero energy modes attached to half-flux tubes. Combined with other results, this allows to explain all strong invariants of two-dimensional topological insulators in terms of a single Fredholm operator.
For the explanation of the quantum Hall effect, Laughlin suggested a Gedankenexperiment during which an extra magnetic flux is inserted adiabatically into a two-dimensional system exposed to a constant magnetic field. This allows to argue for a quantized Hall conductance [Lau]. Actually adiabatics is only needed to establish a connection to the Hall conductance and it is possible to understand the main topological insight of Laughlin’s argument in purely spectral terms, namely as a spectral flow. For example, exactly states flow through the gap above the th Landau level of the Landau operator as a flux is inserted, as can be seen by explicit calculation when modeling the singular flux either by the Aharanov-Bohm gauge or by adequate half-line boundary conditions [AP]. Viewed from the perspective of [ASS] also taken in the present paper, the spectral flow through the th gap of the Landau operator is equal to the Chern number of the associated Fermi projection (on the lowest Landau bands) which in turn can be calculated as the index of the Fredholm operator
where , are the components of the position operator and is the Fermi projection. Indeed, this operator is well-known to be Fredholm if the Fermi level lies in a gap (or even in a region of dynamical localization [BES]), namely its kernel and cokernel are finite dimensional so that its index is well-defined. In this manner, the Laughlin argument appears as a special case of the general connection between the index of a given Fredholm operator and the spectral flow of a wide class of associated unitary dilations, as outlined in Appendix A following Phillips work [Phi] which is also rederived in a companion paper [DS]. Once this perspective is taken, the Laughlin argument acquires a remarkable stability and is not based on any explicit calculation as in [AP]. Here it is presented for gapped tight-binding models with constant magnetic fields and with basically arbitrary hopping elements and potentials (Theorem 1). While such a statement, even in the natural generality presented below, is a folk theorem both in the physics and mathematical physics communities [ASS, BES], a detailed proof does not seem to be in the literature. Closest (but not identical and actually slighly weaker) to ours is a statement in an unpublished manuscript of Macris [Mac], however, there the proof again involves adiabatics which in our opinion is unnatural due to the comments above. Here the general theorem from [Phi, DS] connecting index to spectral flow is applied, and as preparation a careful analysis of the magnetic translations associated to constant magnetic fields perturbed by a flux tube is carried out. These operators lie in a certain extension of the rotation algebra by the compact operators, see Appendix B. It seems perceivable to us that there exists an extension of Theorem 1 to operators with no gap at the Fermi level, but for which the Fermi level lies in a region of dynamical Anderson localization. However, already the statement of such an extension would require a carefully formulated definition of spectral flow (presumably using finite volume approximations) and this goes beyond the scope of this work.
As an application of Theorem 1, a short and intuitive proof of the Elbau-Graf version [EG] of the bulk-edge correspondence is given in Theorem 2 in Section 4. The basic idea of the argument is due to Macris [Mac], but the details in that manuscript are flawed at several crucial points (in particular, the proof of his Lemma 2) and several simplifications are made here (e.g. the gauges for flux tubes are chosen differently). As this argument is based on spectral flow, it is presently not clear how to adapt it for a proof of the bulk-edge correspondence in presence of a mobility gap as stated in [EGS]. It is then shown in Section 5 how Theorem 2 also implies one of the main results of [SKR, KRS] which concerns the bulk-edge correspondence for families of covariant operators. Needless to say, even though these arguments circumvent the use of -theory and cyclic cohomology as explained in [KRS], we believe the -theoretic interpretation to be of great conceptual interest and value.
Section 6 discusses the fate of index of the Fredolm operator and of the Laughlin argument in systems which have supplementary discrete symmetries, namely time reversal symmetry (TRS), particle hole symmetry (PHS) and/or sublattice symmetry (SLS, also called a chiral symmetry). The TRS and PHS can be either even or odd and combinations of all three symmetries lead to the so-called ten universality or symmetry classes which, following Altland and Zirnbauer [AZ], are often labelled by a corresponding Cartan label denoted by CAZ in Table 1. The theory of topological insulators [SRFL, Kit] distinguishes different topological ground states within the CAZ classes. These topological phases are labelled by the so-called strong toplogical invariants (STI) which are usually understood by -theory [Kit, FM, Tan]. As will be discussed below, these -theoretic invariants can take values either in or . For the case of two-dimensional systems these STI are listed in Table 1. In this work, the -theoretic point of view is not further developed, but rather a complementary concrete approach for the labelling of the phases is proposed. Actually, the remarkable fact is that all values of the STI can be computed by analyzing merely one Fredholm operator, namely defined in (1). This is possible because the various physical symmetries lead to symmetries of the Fredholm operator showing that its index is an arbitrary integer in Class A and D, an even integer in Class C, and vanishes in the other class, but has a index as a secondary invariant in Class DIII and AII. To explain this in detail is the object of Section 6. It can be summarized as follows.
Classification Scheme Suppose that the Fermi level lies in a region of dynamical localization in the sense of [BES]. In each of the CAZ classes, the strong invariant of [SRFL, Kit] can be calculated as the index or the -index of the Fredholm operator given in (1). If there is a gap at the Fermi level, all these indices can furthermore be calculated as a spectral flow in the spirit of the Laughlin argument.
Let us give some further explanation as to what the STI actually are in translation invariant and periodic systems, based on [Kit, FM, Tan]. In Class A and AIII, the STI are given by the complex -groups and where is to be interpreted as the two-dimensional momentum space. As the one-point compactification of is the sphere , the group coincides with the reduced -group . In the tight-binding solid state systems analyzed in this work, the sphere should be replaced by the torus and this may (and does in some cases) produce supplementary so-called weak invariants [Kit], which are not analyzed here. These comments transpose verbatim to the remaining cases. There are Real -groups introduced in [Ati] where is the involution induced by for , stemming from complex conjugation in physical space, and . Again . These groups are well-known to be for respectively. By the above classification scheme, these values correspond again precisely to the possible values of the index of . Let us stress though that the above classification scheme based on the invariants of ( and ) applies to systems with broken translation invariance and merely requires dynamical localization which by [BES] assures that is indeed a Fredholm operator. The groups are also the homotopy groups (modulo Bott periodicity) of the classifying spaces for Real -theory, given by skew-adjoint Fredholm operators on a real Hilbert space [AS]. This connection will be further discussed in an up-coming work which will also contain an extension of the classification scheme to other dimensions.
Let us now discuss case by case the invariants of in some more detail, together with the associated physical effects. This list is also a summary of the main results of Section 6.
Class A contains systems without further symmetries and thus, in particular, electronic systems which exhibit a quantum Hall effect (QHE). This is already discussed above. The operator has no particular symmetry and can take arbitrary integer values.
Chiral unitary systems (Class AIII) have a vanishing spectral flow in dimension . Here has vanishing index, and no secondary invariant.
Class D contains Bogoliubov-de Gennes (BdG) Hamiltonians with even PHS, but no further symmetry. This symmetry does not imply any particular symmetry of the Fredholm operator though, and rather connects it to its conjugate Fredholm operator . Hence the spectral flow and can take any integer value. For covariant operators, these integers are equal to the Chern number of the Fermi projection which in turn appear in the Kubo formula for the thermal quantum Hall effect (TQHE) as a prefactor in the Wiedemann-Franz law [VMFT]. Furthermore, in these systems an inserted half-flux quantum is of physical interest as it models a vortex of the pair creation field. The operator at half-flux has again an even PHS. Attached to these vortices are zero modes (ZM) whenever is odd. In second quantization the associated creation operators are self-adjoint so that one also speaks of Majorana modes. While this fact is common knowledge in the physics community [RG], also for tight-binding models [Roy, EF], Theorem 4 seems to provide the first mathematical proof and also establishes the stability of these zero modes for a wide class of operators, containing e.g. random perturbations.
Class C contains BdG Hamiltonians with odd PHS and all the above statements of Class D hold. The physical effect in Class C systems is the spin quantum Hall effect (SQHE) [RG], and is actually equal to the spin Hall conductance as given by the Kubo formula [RG]. The crucial difference w.r.t. Class D is that is always even in Class C systems (Theorem 5). Let us stress that this evenness is not related to the fact that Class C models appear as a pair when obtained as SU-invariant models of a Class D system (as in [AZ]). Our claim is that each of these two Class C models already has an even index, so the even nature is topological as also noted in [SRFL, Kit], see Table 1. This has important implications for the zero modes. Actually, due to the evenness of such zero modes are not stable in Class C, other than claimed in [Roy].
Systems in Class AII have an odd TRS (half-integer spin). In this class, the most prominent toy model with non-trivial topology is the Kane-Mele model [KM], and it has a invariant. The physical effects associated to it are the quantum spin Hall effect (QSHE) and a spin-charge separation (SCS) [QZ, RVL]. It was shown in [SB2] that the odd TRS implies that the Fredholm operator is odd symmetric (in a sense recalled below) and therefore is a well-defined secondary invariant (the index itself vanishes). Indeed, it is shown in Section 6.3 that for the Kane-Mele model. Theorem 6 then shows that such a non-trivial index leads to a characteristic spectral flow, which is intimately related to spin-charge separation [QZ]. This theorem follows from a general result on the spectral flow of dilations of odd symmetric Fredholm operators, proved in [DS] and recalled in Appendix A.
Class DIII comprises models with even PHS and odd TRS. These models inherit from Class AII the possibility to have non-trivial indices. Indeed, it is shown in Section 6.4 how models with such non-trivial topology can be constructed by a doubling procedure, similar as the Kane-Mele model is obtained from two Haldane models. Theorem 7 states that non-trivial topology leads to Kramers’ degenerate double zero modes (DZM) at half flux. In principle, also models in Class CII could have invariants due to the odd TRS, but as the odd PHS already leads to even indices in Class C (Theorem 5), this index is trivial.
In the remaining Classes CI, AI and BDI the even TRS implies that the Fredholm operator is even symmetric in the sense of [SB2] and thus and there is no naturally associated secondary invariant because all Fredholm operators with the corresponding symmetry lie in one connected component (see Theorem 5 in [SB2]).
2 Gauges for flux tubes
The purpose of this section is to write out explicit formulas for two gauges of a flux tube though one cell of the square lattice . One is a discrete version of the standard Aharonov-Bohm gauge, the other one has the vector potential concentrated on a half-line and has already been used in other works [Mac, LZX, EF]. These gauges have different properties which are crucial for the arguments below. As a preparation, some generalities about vector potentials, magnetic fields and gauges on the square lattice are collected in Section 2.1.
2.1 Magnetic potentials, magnetic fields and gauge transformations
Let us view as the vertices of an oriented graph, with oriented edges given by the line segments between nearest vertices. Here , and and denote the two unit vectors of . A magnetic potential on an oriented graph is a real-valued function on the oriented edges, hence in the present case a function satisfying for and . Associated to the magnetic potential is a magnetic field through the cell attached to the upper right at , see Figure 1:
This can be interpreted as the holonomy of along the path around the cell. Only and will be relevant, but it will be convenient to maintain real values. Let us point out that the map is linear, namely . If is a (conventional) vector potential in continuous two-dimensional space, then an associated discretized magnetic potential on in the above sense is obtained by the line integrals:
Any magnetic field can be realized by a magnetic potential and two magnetic potentials realizing the same magnetic field are gauge transformation of each other, as shows the following result.
(i) Given , there exists a magnetic potential such that .
(ii) If and are two magnetic potentials on satisfying , then there exists a so-called gauge transformation such that
Proof. It is known (e.g. [CTT]) that a vector potential can be constructed using a spanning tree for the lattice. For sake of concreteness, let us choose one such tree (see Fig. 1) which then leads to what we call the standard gauge
(ii) Choose as the sum of along a path from to . As , this is independent of the choice of the path.
Next let us introduce the magnetic translations operators and on associated to the magnetic potential :
With given by (2), this is precisely the formula given in [Ara, Theorem 3.2]. From the definition of , it is clear that indeed only the values of are relevant for the magnetic translations. On the other hand, depends on the choice of the magnetic potential and not only on . The following commutation relation states that a phase factor given by the magnetic field is recovered by circulating around one cell:
where denotes the self-adjoint multiplication operator defined by . The second main property of magnetic translations is their behavior under the gauge transformation given in (3):
where denotes again the multiplication operator given by . Another property that is obvious from (4) is its behavior under complex conjugation:
2.2 Some explicit gauges
Let us begin by recalling two standard gauges for a constant magnetic field . The symmetric gauge and Landau gauge are given by
Note that is actually the standard gauge used in Proposition 1 and that the gauge transformation for the difference is given by .
Next let us consider the central object of this work, the discrete flux tube of flux through the cell attached to . The magnetic field of this flux tube is . One possible gauge, termed half-line for sake of concreteness, is
A second gauge is obtained via (2) from the standard singular Aharonov-Bohm gauge in attached at :
Integration then leads to
Using the identity for , it is indeed possible to check that the magnetic field associated with is exactly . Alternatively, one can use the well-known properties of to verify this. The gauge transformation in is explicitly given by
2.3 Magnetic translations with flux tubes
If with magnetic field , the associated (Zak) magnetic translations are denoted by . For so that , we also write instead of . Next let us introduce the magnetic translations with constant magnetic field and flux tube at by
In the following, note that there is a slight modification of the definition of w.r.t. (1).
The operator differences are compact. The operator functions are norm continuous. Furthermore,
where . The commutators are compact operators.
Proof. It follows from (9) that . Thus is a multiplication operator with factor decaying to at infinity and finitely degenerate eigenvalues, so that it is a compact operator. Due to (4) this implies the first claim. As to the second it follows again from the relation (4) and equation (9) which shows that the gauge is linear in with uniformly bounded coefficients, which is sufficient to insure the norm continuity.
To verify (12), let us introduce the gauged magnetic shifts where is the gauge transformation given in (10). Since the -dependance in is given by the exponential in the half-line gauge , one deduces that for all integers . In particular, this implies that
where the unitary , written out using (10), is given by
where the following identities, holding for , were used
For the last claim, let us rewrite using (12)
so that the above allows to conclude.
Appendix B analyzes the C-algebra generated by and . It is an extension of the rotation algebra by the compact operators and this allows to calculate its -theory.
Remark 1 The above proof shows how the unitary depends on the gauge transformation of (10). More generally, let us set
With this notation, the relation between the translations in the half-line gauge and the translations in the Aharonov-Bohm gauge for is given by . From the definition and the explicit form of , it is evident that for all which, in particular, implies . However, a similar relation is not true for . In fact,
is non-vanishing and not even compact, and one has . Hence inserting the flux is not simply implemented by the unitary transformation with , but it really introduces compact perturbations. More precisely, the algebra generated by and is a genuine extension of the rotation algebra (generated by the unperturbed magnetic translations and ) by the compact operators. This is explained in more detail in Appendix B.
Remark 2 The claims of Proposition 2 also hold if is replaced by . On the other hand, replacing by is not allowed because the half-line gauge is actually a non-compact perturbation of the magnetic translations. Let us make this more explicit by analyzing the operator defined by
The difference between and is only in the choice for the gauge of the constant magnetic field , while the gauge for the flux tube is concentrated on the half-line in both cases. Then , but is not compact perturbation of the magnetic translation in the Landau gauge given by . Indeed, the replacement of with in (13) provides
In particular, is not compact. In spite of this unpleasant feature, the half-line gauge is of crucial importance in Section 4.
3 Spectral flow of the Laughlin argument
In this section, Hamiltonians on of the following form will be considered
where (where denotes the ideal of compact operators on and is the unitalization of an algebra obtained by adding multiples of the identity ) and is a uniformly bounded potential and a coupling constant. It will be assumed that the hopping amplitudes decrease sufficiently fast so that
Moreover, for any the conditions
are supposed to hold. They guarantee respectively and .
Remark Definition (15) combined with conditions (17) may seem a little unnatural at first glance. However, Proposition 2 implies that the commutator is a non-vanishing compact operator when , and thus the ordering of the magnetic shifts and becomes relevant. In (15) a particular choice of ordering has been made and this requires (17). Of course, if there are only nearest neighbor hopping terms like in the Harper Hamiltonian this is not an issue. Furthermore, for the commutator is just a number which can be absorbed in .
At , the Hamiltonian is simply denoted by , and for the notations and are used. If with are the only non-vanishing hopping amplitudes, then the Hamiltonian is the two-dimensional representation of the Harper Hamiltonian with constant magnetic flux through each unit cell, and allows, e.g., to add a random potential or a compactly supported scattering-type potential. Furthermore, in the magnetic translations given by (11) add an extra flux through the unit cell attached at .
Let us begin by collecting a few basic mathematical properties of the Hamiltonian (15) which follow rather directly from the properties of the magnetic translations.
Let be continuous. Then the following properties hold:
(i) and are compact.
(iii) with unitary as in (12).
(v) The commutators are compact.
(vi) is norm continuous.
All claims also hold for with .
shows that also is also compact for any and combined with Weierstraß theorem and the norm closedness of the compact operators this implies (i). By Weyl’s theorem also (ii) follows. Furthermore (12) and lead first to , and combined with Weierstraß approximation to (iii). Item (iv) is then a direct consequence of (iii), and (v) combines (iii) and (i):
Finally the continuity (vi) follows from the norm continuity of stated in Proposition 2.
The focus will now be on operators satisfying the following
Gap hypothesis: The Fermi level lies in a spectral gap of and in a gap of the essential spectrum of for all .
The second part of the hypothesis can be slightly weakened by allowing also to depend on , but for sake of simplicity this is not written out here. Let us point out that the Gap hypothesis does not exclude that has bound states close to (namely, discrete spectrum resulting e.g. from a compact potential ). Due to Proposition 3, as function of only the discrete spectrum of is changing and may lead to eigenvalues passing by . In fact, these eigenvalues vary real analytically in due to the analytic dependence of on . The operators at and are isospectral by Proposition 3. Counting the eigenvalue passages along the path weighted by the multiplicities and a positive or negative sign pending on whether the eigenvalues increase or decrease allows to define the integer valued spectral flow by . This is illustrate in Figure 2. As here the eigenvalue curves of the discrete spectrum are real analytic, the intuitive notion of spectral flow indeed leads to mathematically sound definition. Let us note that one may suspect there to be a problem defining the spectral flow in case happens to be an eigenvalue of , but actually there is no issue because is really a closed loop by Proposition 3(iv) so that the flow by is well-defined. A definition of spectral flow for the more general case of norm continuous families of self-adjoint operators can be found in [Phi, DS]. These references also discuss further properties of the spectral flow, such as its homotopy invariance. In order to familiarize the reader with the notion of spectral flow, let us provide an alternative formula which will also be used below.
Suppose that the closed interval lies in a gap of and let be a smooth non-increasing function which is equal to on the left of and on the right of . Then for
Also the r.h.s. is manifestly gauge invariant.
Proof. Let , , denote the finite number of eigenvalues of lying in with normalized eigenvectors where is the maximal number of eigenvalues in for all ’s. These eigenvalues and eigenvectors are real analytic in . As the support of lies in , the operator
is finite rank. By the Fundamental Theorem,
Using one readily concludes the proof. For the final claim, let be the Hamiltonian in another gauge. Then
where in the last equality the cyclicity of the trace is used to cancel out terms.
The following theorem connecting the spectral flow to an index is the central result of this paper. Due to the preparations in Proposition 3, it is a corollary of a general statement of [Phi], also proved in [DS], and recalled in Appendix A.
Suppose the Gap hypothesis holds and let be the Fermi projection of on energies below . Then is a Fredholm operator on and for all its index is given by
Moreover, these expression is constant in .
Proof. By the Gap hypothesis there exists a continuous and non-increasing function such that . Therefore by Proposition 3(v) the operators have compact commutators . This implies the claimed Fredholm property and the constancy of the index on the r.h.s. of (19) follows from the homotopy invariance of the index. Furthermore all the hypothesis of Theorem 8 in Appendix A are verified. Thus
However, the spectral flow on the l.h.s. is precisely equal to the spectral flow in (19).
The following complement to Theorem 1, used in the Section 4 below, shows that one also may cut out finite portions of the physical space without changing the spectral flow. Roughly reformulated, this means that also compactly supported, infinite potentials do not change the spectral flow.
Suppose the Gap hypothesis holds. For set and . Then, for finite and , one has
where is the Fermi projection of . The r.h.s. is still gauge invariant.
Proof. First of all, is again a compact perturbation of so that the essential spectra coincide. Furthermore, the projection commutes with . Now the proof of the first equality is a modification of the proof of Theorem 1 using the homotopy . The second equality follows by the same argument as Proposition 4.
Let us conclude this section by analyzing what happens if several flux tubes are inserted simultaneously.
Suppose the Gap hypothesis holds. Let denote the family of Hamiltonians with a flux through the lattice cells attached to . Then
where is the Fermi projection on energies below lying in the gap.
Proof. Associated to each there is a Dirac phase defined as in formula (12). Setting , one then verifies all the claims of Proposition 3, in particular, the identity as well as compactness of . Now the proof of Theorem 1 shows
Furthermore is compact so that
for some compact operator . The multiplicative property of the index and its invariance under compact perturbations implies
But all of the indices on the r.h.s. are equal to , concluding the proof.
4 Flux tube proof of the bulk-edge correspondence
This section is about the half-space operator acting on simply obtained by restriction from the Hamiltonian given in (15) with and . This corresponds to imposing Dirichlet boundary conditions on the half-plane . In principle all the below also holds for other local boundary conditions, but this is not analyzed in detail (non-local boundary conditions like the spectral boundary conditions of Atiyah-Patodi-Singer are not allowed though). For sake of simplicity, it will be assumed that the sum in (15) is finite by imposing the constraint for some finite range